# 12.7: Summary

In this chapter we have examined several techniques and theorems to assist with the analysis of AC electrical circuits. We began with more practical models for voltage and current sources by adding an internal impedance to set limits on the source's maximum output and make it sensitive to output frequency. For a voltage source, this impedance is in series, its ideal value being a short, just as it was for the DC case. For current sources, the impedance is in parallel, its ideal value being an open.

Source conversions allow us to create an equivalent voltage source for any practical current source and vice versa. An equivalent source is one that will create the same voltage across (and current into) the remaining circuit as did the original source. In some cases, this swap allows multiple sources to be combined into a single source, simplifying analysis. If the associated impedance does not have a zero degree phase angle, then the converted source will not be in phase with the original, but will instead be shifted by the impedance angle.

The superposition theorem states that, for any multi-source linear bilateral network, the contributions of each source may be determined independent of all other sources, the final result being the summation of the contributions. This remains true in the AC case, however, care must be taken regarding phase shifts when combining the various contributions. The original circuit of $$N$$ sources generates $$N$$ new circuits, one for each source under consideration and with all other sources replaced by their ideal internal impedance.

Thévenin's and Norton's theorems allow the simplification of complex linear single port (i.e., two connecting points) networks. The AC Thévenin equivalent consists of a voltage source with a series impedance while the AC Norton equivalent consists of a current source with a parallel impedance. These impedances can be represented in general as a resistance in series with a reactance, and given an operating frequency, the reactance can be turned into a capacitance or inductance. These equivalents, when replacing the original sub-circuit, will create the same voltage across the remainder of the circuit with the same current draw. In other words, the remainder of the circuit will see no difference between being driven by the original sub-circuit or by either the Thévenin or Norton equivalents.

The maximum power transfer theorem states that for a simple voltage source with an internal impedance driving a simple load, the maximum load power will be achieved when the load impedance equals the complex conjugate of the internal impedance. The complex conjugate has the same real or resistive value, however, the reactive portion is of the opposite sign. This results in a cancellation of the reactive components, leaving just the resistive portions and maximizing load current. At this point, efficiency will be 50%. If the load impedance is higher than the internal impedance, the load power will not be as great, however, the system efficiency may improve, depending on the phase angle.

Delta-Y conversions allow the generation of equivalent “three connection point” impedance networks. RLC networks with three elements in the shape of a triangle or delta (with one connection point at each corner) may be converted into a three element network in the shape of a Y or T, or vice versa. The two versions will behave identically to the remainder of the circuit. This allows the simplification of some circuits and eases analysis.

## Review Questions

1. What are the ideal internal impedances of AC voltage and current sources?

2. Outline the process of converting an AC voltage source into an AC current source, and vice versa.

3. In general, describe the process of using superposition to analyze a multisource circuit.

4. What do Thévenin's and Norton's theorems state? How are they related?

5. What are the conditions to achieve maximum power transfer for AC circuits? How does this differ from the DC version?

6. What are delta and Y configurations? How are they related?